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Restricting a function defined on an étale groupoid to an isotropy group

Let $\mathcal G$ be an étale groupoid, let $x$ be a point in the unit space of $\mathcal G$, and let $\mathcal G(x)$ be the isotropy group of $x$. If $f$ is a continuous, complex valued, compactly ...
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Is it possible to characterize the elements of the C$^*$-algebra of an open subgroupoid?

$\newcommand{\Cstar}{C^*_{\text{red}}}\newcommand{\G}{\mathscr G}\newcommand{\H}{\mathscr H}$Let $\G$ be an etale groupoid, let $U$ be an open subset of $\G^{(0)}$, and let $$ \H = \{\gamma \in \G:...
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